Extremes of a certain class of Gaussian processes

Abstract

We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend − ct β for some constants c, β>0 and a variance t 2 H . We derive the tail behaviour of these extremes and show that they occur mainly in the neighbourhood of the unique point t 0 where the related boundary function ( u+ ct β )/ t H is minimal. We consider the case that H< β.